Generalised vectorial ∞-eigenvalue nonlinear problems for L∞ functionals. (June 2022)

• Record Type: Journal Article
• Authors: Katzourakis, Nikos
• Is Part Of: Nonlinear analysis. Volume 219(2022)
• Publication Date: 2022-06
• Subjects: 35D30 -- 35D40 -- 35J47 -- 35J92 -- 35J70 -- 35J99 -- 35P30; ∞-Eigenvalue problem -- Nonlinear Eigenvalue problems -- ∞-Laplacian -- L∞ functionals -- Absolute minimisers -- Calculus of variations in L∞ -- Lagrange multipliers; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112806
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let Ω ⋐ R n, f ∈ C 1 ( R N × n ) and g ∈ C 1 ( R N ), where N, n ∈ N . We study the minimisation problem of finding u ∈ W 0 1, ∞ ( Ω ; R N ) that satisfies ‖ f ( D u ) ‖ L ∞ ( Ω ) = inf { ‖ f ( D v ) ‖ L ∞ ( Ω ) : v ∈ W 0 1, ∞ ( Ω ; R N ), ‖ g ( v ) ‖ L ∞ ( Ω ) = 1 }, under natural assumptions on f, g . This includes the ∞ -eigenvalue problem as a special case. Herein we prove the existence of a minimiser u ∞ with extra properties, derived as the limit of minimisers of approximating constrained L p problems as p → ∞ . A central contribution and novelty of this work is that u ∞ is shown to solve a divergence PDE with measure coefficients, whose leading term is a divergence counterpart equation of the non-divergence ∞ -Laplacian. Our results are new even in the scalar case of the ∞ -eigenvalue problem.